Alexander Invariant

The Alexander invariant $H_*(\tilde X)$ of a Knot $K$ is the Homology of the Infinite cyclic cover of the complement of $K$, considered as a Module over $\Lambda$, the Ring of integral Laurent Polynomials. The Alexander invariant for a classical Tame Knot is finitely presentable, and only $H_1$ is significant.

For any Knot $K^n$ in ${\Bbb{S}}^{n+2}$ whose complement has the homotopy type of a Finite Complex, the Alexander invariant is finitely generated and therefore finitely presentable. Because the Alexander invariant of a Tame Knot in ${\Bbb{S}}^3$ has a Square presentation Matrix, its Alexander Ideal is Principal and it has an Alexander Polynomial denoted $\Delta(t)$.

See also Alexander Ideal, Alexander Matrix, Alexander Polynomial

References

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 206-207, 1976.